March 21, 2022

Competition

an interaction between organisms (intraspecific) or between species (interspecific) in which fitness of one is lowered by the presence of another.

We’ve already talked about intraspecific competition!

Fitness is Reproductive Success

  • survive to reproductive age
  • find a mate
  • produce (raise) offspring

Competitive Exclusion Principle

If two species are using the same resource and share the same niche, they can’t coexist:

Georgiy Franzevich Gause (1910-1986) Ѓeоргий Францевич Гаузе

Competitive Exclusion Principle

But … coexistence!?

MacArthur’s warblers

Five species …. all insectivorous, all share the same set of boreal conifers in homogeneous stands. In principle (specifically - according to the competitive exclusion principle), no TWO species should co-exist.

How do FIVE coexist?

Resource partitioning!

Spatial niche partitioning ….

Temporal niche partitioning ….

Temporal niche partitioning ….


LaPoint and Gurarie (in progress)

Apparent competition

Species A eats Species B and C, if Species B increases, Species C is in trouble.

Major habitat fragmentation from oil-gas extraction.

Serrouya et al. (2017)

What can models tell us about coexistence and competition?

Remember Logistic Growth

Two species, \(N_1\) and \(N_2\).

\[{dN_1 \over dt} = r_1 N_1 \left(1-{N_1 \over K_1}\right)\]

\[{dN_2 \over dt} = r_2 N_2 \left(1-{N_2 \over K_2}\right)\]

each with its own \(r\) and \(K\).

Intra-specific competition: Lotka-Volterra model

Two species, \(N_1\) and \(N_2\).

\[{dN_1 \over dt} = r_1 N \left(1-{N_1 \over K_1} - \alpha {N_2\over K_1}\right)\]

\[{dN_2 \over dt} = r_2 N \left(1-{N_2 \over K_2} - \beta { N_1\over K_2}\right)\]

  • \(\alpha\) - effect of \(M\) on \(N\) … i.e. how many \(N_1\)’s does a single \(N_2\) cut things down by.
  • \(\beta\) - effect of \(N\) on \(M\).

Squirlicorn vs. Pegamunk

Limited space, Limited carrying capacity, Mutual animosity (periodic skewering and dropping on rocks) …. can they get along?

Consider Species 1

Population 1 size not changing.

\[0 = r_1 N_1 \left(1 - {N_1 \over K_1} - {\alpha N_2 \over K_1}\right)\] boils down (quickly) to

\[N_1 = K_1 - \alpha N_2\]

Isocline of Zero Growth where \({dN_1 \over dt} = 0\)

How does a zero-growth isocline work?

  • Above the line, the numbers decrease
  • Below the line, the numbers increase

Add Species 2

Population 2 size not changing.

\[0 = r_2 N_2 \left(1 - {N_2 \over K_2} - {\beta N_1 \over K_2}\right)\]

\[N_2 = K_2 - \beta N_1\]

Isocline of Zero Growth where \({dN_2 \over dt} = 0\)

Follow the arrows ….

To see species 1 drive species 2 to extinction.

Don’t have to be parallel

What about a criss-cross?

If \(K_2 > {K_1 \over \alpha}\) and \(K_1 > {K_2 \over \beta}\)

either Species 1 or Species 2 will go extinct.

What about a criss-cross?

BUT if: \(K_2 > {K_1 \over \alpha}\) and \(K_1 > {K_2 \over \beta}\)

There will be coexistence!

At an equilibrium that is (of course) smaller than either \(K_1\) or \(K_2\)

Competition ….

\[{dN_1 \over dt} = r_1 N \left(1-{N_1 \over K_1} - \alpha {N_2 \over K_1}\right)\]

\[{dN_2 \over dt} = r_2 N \left(1-{N_2 \over K_2} - \beta {N_1 \over K_2}\right)\]

is the outcome of 2 density dependent phenomena:

  1. Intra-specific competition (\(K_1\) and \(K_2\))
  2. Inter-specific competition (\(\alpha\) and \(\beta\))

Patterns depend on the scale and particulars of the competition intensity.

Focus on the the isoclines of zero growth

\[N_2 = {K_1 \over \alpha} - {1\over \alpha} N_1\]

\[N_2 = K_2 - \beta N_1\]

Extinction vs. Equilibria

If:

  • \({1\over\beta} < {K_1 \over K_2} > \alpha\); \(N_1\) wins
  • \({1\over\beta} > {K_1 \over K_2} < \alpha\); \(N_2\) wins
  • \({1\over\beta} < {K_1 \over K_2} < \alpha\); \(N_1\) or \(N_2\) wins
  • \({1\over\beta} > {K_1 \over K_2} > \alpha\); \(N_1\) and \(N_2\) coexist

Equilibria (stable or not) at:

  • \(N^*_1 = {K_1 - \alpha K_2 \over 1-\alpha\beta}\)
  • \(N^*_2 = {K_2 - \beta K_1 \over 1-\alpha\beta}\)

Co-existence essentially occurs when ….

Intra-specific > Inter-specific competition

\[{1\over\beta} > {K_1 \over K_2} > \alpha\]

\(\beta\) and \(\alpha\) small, in a very particular way relative to the ratio of carrying capacities.

The theorists ….

Vito Volterra (1860-1940)

Mathematician.

Father-in-law of fisheries biologist.

One of 12 (out of 1250) of Italian professors to not sign an oath of loyalty to Mussolini.

Alfred J. Lotka (1880-1949)

1860-1940

Chief statisticitan at Met Life Insurance.

Independently published these equations, with more or less this application, in 1926 and 1925.

Nice theory you’ve go there but …..

Detecting and quantifying competition in anything resembling a real system is very difficult!

Gerbil competition experiment

Gerbil coexistence and habitat partitioning

  • Zone I, both species prefer the semistabilized dune.
  • Zone II G. pyramidum prefers the semistabilized dune and G. allenbyi exhibits random utilization of both habitats
  • Zone III G. allenbyi exhibits preference for the stabilized sand and G. pyramidum prefers semistabilized dune.
  • Zone IV G. allenbyi prefers stabilized sand and G. pyramidum selects semistabilized dune but also uses the stabilized sand.

Ultimately ….

Gerbils CAN get along. But the isoclines are compilated!